Whilst unfortunately, Bullhorn Support are not able to directly torubleshoot your PHP code, I can recommend some suggestions which will hopefully help moving forward.

Firstly, I've successfully made this call using Postman (https://www.getpostman.com/) which is our recommended API test platform. A screenshot of my successful attempt can be found below:

I achieved this by setting the body to form-data, and then declaring the multipart/form-data as the chosen file. Once you've run this in postman, you can press the "code" button in the top right (highlighted in the above image) which will let you view the code needed to reproduce this functionality in different languages, including PHP with cURL. If you can action this on your side, this will hopefully give you the steps needed to complete this on your side.